The Power of Interdisciplinary Bridges:

Throwing the Net Widely

Deborah Hughes HallettUniversity of Arizona & Harvard Kennedy School

Common Core Standards• Focus on mathematical structure and understanding

What does this mean for teachers?• Need deeper understanding of the mathematics

– Theoretical: How topics fit together– Applied: How and where this is used

Teachers need a “view from the balcony” of the mathematics they teach and where it is leading

What are Students’ Views?• Some students appreciate mathematical elegance• Some students absorb an idea better when they know

how it is used

How Do Students’ Views Affect Teachers’ Needs?

• Teachers may not know where a topic they teach is leading, or why it is important

• Mathematicians can provide professional development that fills in these gaps and enables teachers “to climb to the balcony”

What Kind of Professional Development is Needed?

• Interaction between teachers, mathematicians, mathematics educators

• Providing perspective and inspiration• Answering questions, creating a sense of community• Providing continuity of expectations as students

move between school and college

We will look at examples that respond to “How is this used?”

“Why Do I Need to Know This?”: Contexts for Rates of Change

• Slope• Linear and exponential growth • Change and percent change• Rate of change and percent rate of change• The number e• Continuous and annual rate of change• Extrapolation and interpolation

Understanding the Debate about Climate

Climate change in past:• Temperatures rose 1-2 F during 1906-2005• CO2 in atmosphere increased from 316 ppm in 1959 to 386 ppm

in 2008• Greenhouse gasses tend to warm the planet From Tom Pfaff and http://www.epa.gov/climatechange/science/stateofknowledge.html and http://www.esrl.noaa.gov/gmd/ccgg/trends/index.html#mlo

Climate change predictions:• Temperatures will continue to rise, perhaps accelerating• Greenhouse gasses such as CO2 in atmosphere will increase,

perhaps accelerating• Sea levels will rise, leading to flooding • From http://www.esrl.noaa.gov/gmd/ccgg/trends/ and http://data.giss.nasa.gov/gistemp/tabledata/GLB.Ts.txt

1950 1960 1970 1980 1990 2000 2010300

310

320

330

340

350

360

370

380

f(x) = 1.37118445322793 x − 2374.65840184453R² = 0.989305896183111

Average CO2, Mauna Loa, Hawaii

Year

CO2,

par

ts p

er m

illio

n (p

pm)

Predicting Future Carbon Dioxide Levels

1950 1960 1970 1980 1990 2000 2010300310320330340350360370380

f(x) = 0.123016827118996 exp( 0.00400195852798679 x )R² = 0.992818679258666

Average CO2, Mauna Loa, Hawaii

Year

CO2,

par

ts p

er m

illio

n (p

pm)

http://cdiac.ornl.gov/new/keel_page.html

“Why Do I Need to Know This?”: Contexts for Algebraic Manipulation

• Linear functions • Logarithms• The number e• Linearization

y = 0.0325t + 2.1214

0

2

4

6

0 50 100

t, years since 1913

y, ln

(CPI

)Linearization in Economics

How fast has CPI (Consumer Price Index) grown over last century?

Another View of Same CPI Data

CPI, with 1982-84 defined to be 100

y = 8.3132e0.0326x

R2 = 0.9185

0

50

100

150

200

250

0 20 40 60 80 100

Years since 1913

CPI

Linearization in Biology

• V0 initial velocity of reaction• [S]0 is initial concentration of substrate• Vmax, KM are constants

0

0max0 ][

][SKSV

Vm

Michaelis-Menten Equation

How Do You Know If a Reaction Follows Michaelis-Menten?

Linearize:

max0max0

0max

0

0

1][

11

][][1

VSVK

V

SVSK

V

M

M

“Why Do I Need to Know This?”: Contexts for Proportion & Probability

• Ratio and proportion• Rounding and estimation• Probability• Conditional probability (what is in the

denominator?)• Bayes’ Theorem

Is There Racial Profiling in LA?From Lily Khadjavi, Loyola Marymount

LAPD motor vehicle stops resulting in search (01/01/04-06/30/04)

Race of driver

White African-American

Search 5849 9882 15731

No search 106104 49439 155543

Numbers in Thousands

Race of driver

White African-American

Search 6 10 16

No search 110 50 160

•About 5% of white drivers were searched when stopped•About 20% of African-American drivers were searched when stopped

Mathematics: Estimation and Percentage

Another Example:“Drug Testing Set For Student Athletes In Illinois”

Reporting Dorothy Tucker CHICAGO (CBS) Jan 15, 2008 6:23 pm US/Central

As CBS 2's Dorothy Tucker explains, now some high school athletes in Illinois will have to take another [test]…, a drug test.

This year's state finals will be the last games students play without being tested for steroids.

Next year, whether it's basketball, football, or any other sport, whether it's girls or boys, every athlete competing in sectionals, regionals or championship games will be subject to random testing of banned drugs.

Questions:How likely is it that a student is falsely accused? How likely is it that a cheating student is missed?

“Why Do I Need to Know This?”: Contexts for Graph Reading & Calculus

• Graphical differentiation• Differential equations

– Slope fields / Phase diagrams– Separation of variables– Integration techniques: partial fractions

• Logistic function

– Estimating parameters

– Fitting lines to data

Predicting Peak Oil Production

What is peak oil production? When will/did US oil production peak?When will/did world oil production peak?

Why is this important? What are the economic implications?

(Analyze using 2008 data at www.eia.doe.gov. Calculus, Hughes Hallett at al)

0 50000000 100000000 150000000 200000000 2500000000

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

dP/dt vs. PP is total oil produced in US since start of time

P

dP/d

t

US Oil Production (1900 to 2005)

0 20 40 60 80 100 1200

50000000

100000000

150000000

200000000

250000000

P vs. tHubbert assumed approximate limit of 200,000,000

thousand barrels

Year (t1900 = 0)

P (T

otal

Oil

Pum

ped

in 1

000'

s of

Bar

rels

)

Total US Oil Pumped since the Beginning of Time

http://en.wikipedia.org/wiki/Predicting_the_timing_of_peak_oil

US oil production (crude oil only)

and Hubbert high estimate, peak ≈

1970

Norway, peak ≈ 2001

Preventing the Spread of SARS

There is an outbreak of SARS in a nearby city. As you are working in the mayor’s office, you are asked which is the most effective policy for preventing the spread to the disease to your city:

I. Close off the city from contact with the infected region. Shut down roads, airports, trains, busses, and other forms of direct contact.

II. Install a quarantine policy. Isolate anyone who has been in contact with a SARS patient or anyone who shows symptoms of SARS.

(Analyze using 2003 World Health Organization data from Hong Kong. Calculus, Hughes Hallett at al)

Progress of the Disease: Without Quarantine

Progress of the Disease: With Quarantine

Now is The Time

• Common Core Standards have led to calls for more and better professional development for teachers

• What are our own departments/states doing?• What could we be doing?• Interaction between teachers and

mathematicians is valuable : We can contribute a “view from the balcony”